[Paper Review] Piecewise constructions of inverses of cyclotomic mapping permutations

Yanbin Zheng, Yuyin Yu+1 more|arXiv (Cornell University)|Oct 14, 2015

Coding theory and cryptography3 citations

TL;DR

This paper introduces a piecewise interpolation formula that generalizes Lagrange interpolation to efficiently compute explicit inverses of permutation polynomials over arbitrary finite fields, particularly for cyclotomic mapping permutations and polynomials of the form $x^r h(x^{(q-1)/d})$. The method enables closed-form inversion even for large $q$, where traditional methods become computationally infeasible.

ABSTRACT

Given a permutation polynomial $f(x)$ of a small finite field $\mathbb{F}_q$, the inverse $f^{-1}(x)$ over $\mathbb{F}_q$ could be determined by using the Lagrange Interpolation Formula. But for a large $q$, finding the explicit expression of $f^{-1}(x)$ is usually a hard problem. A piecewise interpolation formula for the inverses of permutation polynomials of arbitrary finite fields is introduced, which extends the Lagrange Interpolation Formula. The explicit inverses of cyclotomic mapping permutations are constructed to demonstrate the formula. The explicit inverses of permutation polynomials of the form $x^rh(x^{(q-1)/d})$ are presented.

Motivation & Objective

• To address the challenge of computing explicit inverses of permutation polynomials over large finite fields, where standard methods like Lagrange interpolation become impractical.
• To generalize the Lagrange interpolation formula to a piecewise version that applies to arbitrary finite fields and complex polynomial structures.
• To construct explicit inverse expressions for permutation polynomials of the form $x^r h(x^{(q-1)/d})$ using cyclotomic mapping properties.
• To provide a systematic framework for inverting permutation polynomials in cryptographic and coding-theoretic applications requiring efficient inversion.
• To demonstrate the feasibility and efficiency of the proposed method through explicit constructions on cyclotomic mapping permutations.

Proposed method

• Proposes a piecewise interpolation formula that partitions the field $ackslash mathbb{F}_q$ into cyclotomic classes based on the multiplicative structure of $q-1$.
• Uses the algebraic structure of cyclotomic mapping permutations to define interpolation segments over each coset of the subgroup of $d$-th powers in $ackslash mathbb{F}_q^\times$.
• Applies a modified interpolation scheme within each cyclotomic class, leveraging the functional form $f(x) = x^r h(x^{(q-1)/d})$ to derive local inverse expressions.
• Combines local inverse polynomials from each class into a global piecewise-defined inverse function $f^{-1}(x)$ over $ackslash mathbb{F}_q$.
• Employs properties of roots of unity and multiplicative characters to ensure consistency and correctness of the piecewise inverse across field elements.
• Validates the method by constructing explicit inverses for known classes of permutation polynomials, including those with $d$ dividing $q-1$.

Experimental results

Research questions

• RQ1 How can the inverse of a permutation polynomial over a large finite field be computed efficiently when standard interpolation becomes infeasible?
• RQ2 Can a generalized interpolation method be developed that applies to arbitrary finite fields and complex polynomial forms like $x^r h(x^{(q-1)/d})$?
• RQ3 What structural properties of cyclotomic mapping permutations enable piecewise inversion, and how can they be exploited algorithmically?
• RQ4 To what extent does the piecewise interpolation formula reduce computational complexity compared to full Lagrange interpolation?
• RQ5 Can explicit inverse expressions be derived for permutation polynomials of the form $x^r h(x^{(q-1)/d})$ using this method?

Key findings

• The proposed piecewise interpolation formula successfully generalizes Lagrange interpolation to arbitrary finite fields and enables explicit inversion of permutation polynomials that are otherwise difficult to invert.
• The method allows for the construction of closed-form inverse expressions for permutation polynomials of the form $x^r h(x^{(q-1)/d})$ by leveraging cyclotomic class decomposition.
• Explicit inverses are derived for cyclotomic mapping permutations, demonstrating the method's applicability to important classes of permutation polynomials.
• The piecewise approach reduces computational complexity compared to full Lagrange interpolation, especially for large $q$, by working within smaller, structured field subsets.
• The framework provides a systematic and algebraically sound method to compute inverses without requiring exhaustive search or numerical approximation.
• The results confirm that the structure of $x^r h(x^{(q-1)/d})$ polynomials allows for efficient inversion through localized interpolation over cyclotomic cosets.